Properties

Label 175.5.g.a
Level $175$
Weight $5$
Character orbit 175.g
Analytic conductor $18.090$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,5,Mod(43,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.43");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 175.g (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.0897435397\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 6 \beta_{3} q^{3} - 9 \beta_{2} q^{4} + 42 q^{6} + 7 \beta_1 q^{7} - 25 \beta_{3} q^{8} - 171 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 6 \beta_{3} q^{3} - 9 \beta_{2} q^{4} + 42 q^{6} + 7 \beta_1 q^{7} - 25 \beta_{3} q^{8} - 171 \beta_{2} q^{9} + 102 q^{11} - 54 \beta_1 q^{12} + 64 \beta_{3} q^{13} + 49 \beta_{2} q^{14} + 31 q^{16} + 46 \beta_1 q^{17} - 171 \beta_{3} q^{18} - 350 \beta_{2} q^{19} + 294 q^{21} + 102 \beta_1 q^{22} - 96 \beta_{3} q^{23} - 1050 \beta_{2} q^{24} - 448 q^{26} - 540 \beta_1 q^{27} - 63 \beta_{3} q^{28} + 1510 \beta_{2} q^{29} - 1708 q^{31} - 369 \beta_1 q^{32} - 612 \beta_{3} q^{33} + 322 \beta_{2} q^{34} - 1539 q^{36} - 704 \beta_1 q^{37} - 350 \beta_{3} q^{38} + 2688 \beta_{2} q^{39} + 1022 q^{41} + 294 \beta_1 q^{42} - 796 \beta_{3} q^{43} - 918 \beta_{2} q^{44} + 672 q^{46} + 1376 \beta_1 q^{47} - 186 \beta_{3} q^{48} + 343 \beta_{2} q^{49} + 1932 q^{51} + 576 \beta_1 q^{52} - 536 \beta_{3} q^{53} - 3780 \beta_{2} q^{54} + 1225 q^{56} - 2100 \beta_1 q^{57} + 1510 \beta_{3} q^{58} + 5530 \beta_{2} q^{59} + 5572 q^{61} - 1708 \beta_1 q^{62} - 1197 \beta_{3} q^{63} - 3079 \beta_{2} q^{64} + 4284 q^{66} + 2456 \beta_1 q^{67} - 414 \beta_{3} q^{68} - 4032 \beta_{2} q^{69} - 898 q^{71} - 4275 \beta_1 q^{72} - 746 \beta_{3} q^{73} - 4928 \beta_{2} q^{74} - 3150 q^{76} + 714 \beta_1 q^{77} + 2688 \beta_{3} q^{78} + 9010 \beta_{2} q^{79} - 8829 q^{81} + 1022 \beta_1 q^{82} + 734 \beta_{3} q^{83} - 2646 \beta_{2} q^{84} + 5572 q^{86} + 9060 \beta_1 q^{87} - 2550 \beta_{3} q^{88} - 2450 \beta_{2} q^{89} - 3136 q^{91} - 864 \beta_1 q^{92} + 10248 \beta_{3} q^{93} + 9632 \beta_{2} q^{94} - 15498 q^{96} + 406 \beta_1 q^{97} + 343 \beta_{3} q^{98} - 17442 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 168 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 168 q^{6} + 408 q^{11} + 124 q^{16} + 1176 q^{21} - 1792 q^{26} - 6832 q^{31} - 6156 q^{36} + 4088 q^{41} + 2688 q^{46} + 7728 q^{51} + 4900 q^{56} + 22288 q^{61} + 17136 q^{66} - 3592 q^{71} - 12600 q^{76} - 35316 q^{81} + 22288 q^{86} - 12544 q^{91} - 61992 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.87083 + 1.87083i
1.87083 1.87083i
−1.87083 1.87083i
1.87083 + 1.87083i
−1.87083 + 1.87083i −11.2250 11.2250i 9.00000i 0 42.0000 −13.0958 + 13.0958i −46.7707 46.7707i 171.000i 0
43.2 1.87083 1.87083i 11.2250 + 11.2250i 9.00000i 0 42.0000 13.0958 13.0958i 46.7707 + 46.7707i 171.000i 0
57.1 −1.87083 1.87083i −11.2250 + 11.2250i 9.00000i 0 42.0000 −13.0958 13.0958i −46.7707 + 46.7707i 171.000i 0
57.2 1.87083 + 1.87083i 11.2250 11.2250i 9.00000i 0 42.0000 13.0958 + 13.0958i 46.7707 46.7707i 171.000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.5.g.a 4
5.b even 2 1 inner 175.5.g.a 4
5.c odd 4 2 inner 175.5.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.5.g.a 4 1.a even 1 1 trivial
175.5.g.a 4 5.b even 2 1 inner
175.5.g.a 4 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 49 \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 49 \) Copy content Toggle raw display
$3$ \( T^{4} + 63504 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 117649 \) Copy content Toggle raw display
$11$ \( (T - 102)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 822083584 \) Copy content Toggle raw display
$17$ \( T^{4} + 219395344 \) Copy content Toggle raw display
$19$ \( (T^{2} + 122500)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 4161798144 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2280100)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1708)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 12036125753344 \) Copy content Toggle raw display
$41$ \( (T - 1022)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 19671992537344 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 175658761191424 \) Copy content Toggle raw display
$53$ \( T^{4} + 4044410589184 \) Copy content Toggle raw display
$59$ \( (T^{2} + 30580900)^{2} \) Copy content Toggle raw display
$61$ \( (T - 5572)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 17\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T + 898)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 15175792854544 \) Copy content Toggle raw display
$79$ \( (T^{2} + 81180100)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 14222643349264 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6002500)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1331374437904 \) Copy content Toggle raw display
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